Probabilistic Time

2018

Nuclear Physics B

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For "static memory materials" the bulk properties depend on boundary conditions. Such materials can be realized by classical statistical systems which admit no unique equilibrium state. We describe the propagation of information from the boundary to the bulk by classical wave functions. The dependence of wave functions on the location of hypersurfaces in the bulk is governed by a linear evolution equation that can be viewed as a generalized Schrödinger equation. Classical wave functions obey the superposition principle, with local probabilities realized as bilinears of wave functions. For static memory materials the evolution within a subsector is unitary, as characteristic for the time evolution in quantum mechanics. The space-dependence in static memory materials can be used as an analogue representation of the time evolution in quantum mechanics -such materials are "quantum simulators". For example, an asymmetric Ising model on a Euclidean two-dimensional lattice represents the time evolution of free relativistic fermions in two-dimensional Minkowski space. Ising models for massless relativistic fermions 227.

Section: Introductionmentioning

confidence: 99%

Information transport in classical statistical systems

2018

Nuclear Physics B

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“…Once one realizes that quantum systems can be implemented as suitable subsystems of classical statistical systems [51], several interesting questions emerge. Is time an ordering concept [49] in a general probabilistic ensemble covering the universe from the "infinite past" to the "infinite future"? Why and how do the quantum conditions necessary for the realization of quantum subsystems arise in the ubiquitous way observed in nature?…”

Section: Discussionmentioning

confidence: 99%

Quantum computing with classical bits

2019

Nuclear Physics B

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A bit-quantum map relates probabilistic information for Ising spins or classical bits to quantum spins or qubits. Quantum systems are subsystems of classical statistical systems. The Ising spins can represent macroscopic two-level observables, and the quantum subsystem employs suitable expectation values and correlations. We discuss static memory materials based on Ising spins for which boundary information can be transported through the bulk in a generalized equilibrium state. They can realize quantum operations as the Hadamard or CNOT-gate for the quantum subsystem. Classical probabilistic systems can account for the entanglement of quantum spins. An arbitrary unitary evolution for an arbitrary number of quantum spins can be described by static memory materials for an infinite number of Ising spins which may, in turn, correspond to continuous variables or fields. We discuss discrete subsets of unitary operations realized by a finite number of Ising spins. They may be useful for new computational structures. We suggest that features of quantum computation or more general probabilistic computation may be realized by neural networks, neuromorphic computing or perhaps even the brain. This does neither require small isolated entities nor very low temperature. In these systems the processing of probabilistic information can be more general than for static memory materials. We propose a general formalism for probabilistic computing for which deterministic computing and quantum computing are special limiting cases.

“…(25) becomes a linear differential equation, generalizing the Schrödinger equation in quantum mechanics. Similarly, the conjugate wave function obeys ∂ tq (t) = −W T (t)q(t), (27) whereW (t) = 1 2ǫ S(t − ǫ) − S −1 (t) (28) equals W (t) if S is independent of t. Without loss of generality, we can formally write…”

Section: Quantum Formalismmentioning

confidence: 99%

“…In classical statistics the t-derivativeȦ of a local observable A is typically also a local observable. The operators for A andȦ do not commute, similar to quantum mechanics [27]. Let us consider a classical local observable A(t).…”

Section: Derivative Observablesmentioning

confidence: 99%

“…For arbitrary S ρτ (t) they may no longer correspond to overall classical probability distributions. We can define "strong local equivalence classes" of all w[n] that lead to the same density matrix ρ ′ (t) for all t. (A different and weaker notion of equivalence requires only the same local probability distribution p τ (t) [16,27].) Since expectation values and correlations of local observables are completely determined by ρ ′ (t), all members of a given equivalence class will lead to the same local physics.…”

Section: Strong Local Equivalence Classesmentioning

confidence: 99%

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Quantum formalism for classical statistics

2018

Annals of Physics

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In static classical statistical systems the problem of information transport from a boundary to the bulk finds a simple description in terms of wave functions or density matrices. While the transfer matrix formalism is a type of Heisenberg picture for this problem, we develop here the associated Schrödinger picture that keeps track of the local probabilistic information. The transport of the probabilistic information between neighboring hypersurfaces obeys a linear evolution equation, and therefore the superposition principle for the possible solutions. Operators are associated to local observables, with rules for the computation of expectation values similar to quantum mechanics. We discuss how non-commutativity naturally arises in this setting. Also other features characteristic of quantum mechanics, such as complex structure, change of basis or symmetry transformations, can be found in classical statistics once formulated in terms of wave functions or density matrices. We construct for every quantum system an equivalent classical statistical system, such that time in quantum mechanics corresponds to the location of hypersurfaces in the classical probabilistic ensemble. For suitable choices of local observables in the classical statistical system one can, in principle, compute all expectation values and correlations of observables in the quantum system from the local probabilistic information of the associated classical statistical system. Realizing a static memory material as a quantum simulator for a given quantum system is not a matter of principle, but rather of practical simplicity. 45 11.3. Strong local equivalence classes . . . . . 45 12.Connections between classical and quantum statistics 46 12.